Orthogonal projection
inside Petrie polygon
The central orange vertex is doubled
TypeRegular 7-polytope
Schläfli symbol {4,35}
Coxeter-Dynkin diagrams

6-faces14 {4,34}
5-faces84 {4,33}
4-faces280 {4,3,3}
Cells560 {4,3}
Faces672 {4}
Vertex figure6-simplex
Petrie polygontetradecagon
Coxeter groupC7, [35,4]

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.

Related polytopes

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes.

Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces.

Cartesian coordinates

Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are


while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6) with -1 < xi < 1.


orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Dihedral symmetry [6] [4]


This hypercube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:7:21:35:35:21:7:1.

Petrie polygon, skew orthographic projection

Another orthogonal projection

Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.


External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
This article is issued from Wikipedia - version of the 4/17/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.